On the stochastic Cahn-Hilliard equation with a singular double-well potential

We prove well-posedness and regularity for the stochastic pure Cahn-Hilliard equation under homogeneous Neumann boundary conditions, with both additive and multiplicative Wiener noise. In contrast with great part of the literature, the double-well potential is treated as generally as possible, its convex part being associated to a multivalued maximal monotone graph everywhere defined on the real line on which no growth nor smoothness assumptions are assumed. The regularity result allows to give appropriate sense to the chemical potential and to write a natural variational formulation of the problem. The proofs are based on suitable monotonicity and compactness arguments in a generalized variational framework.Comment: 37 page

A doubly nonlinear evolution problem related to a model for microwave heating

This paper is concerned with the existence and uniqueness of the solution to a doubly nonlinear parabolic problem which arises directly from a circuit model of microwave heating. Beyond the relevance from a physical point of view, the problem is very interesting also in a mathematical approach: in fact, it consists of a nonlinear partial differential equation with a further nonlinearity in the boundary condition. Actually, we are going to prove a general result: the two nonlinearities are allowed to be maximal monotone operators and then an existence result will be shown for the resulting problem.Comment: Key words and phrases: nonlinear parabolic equation, nonlinear boundary condition, existence of solution

Optimal distributed control of a stochastic Cahn-Hilliard equation

We study an optimal distributed control problem associated to a stochastic Cahn-Hilliard equation with a classical double-well potential and Wiener multiplicative noise, where the control is represented by a source-term in the definition of the chemical potential. By means of probabilistic and analytical compactness arguments, existence of an optimal control is proved. Then the linearized system and the corresponding backward adjoint system are analysed through monotonicity and compactness arguments, and first-order necessary conditions for optimality are proved.Comment: Key words and phrases: stochastic Cahn-Hilliard equation, phase separation, optimal control, linearized state system, adjoint state system, first-order optimality condition

Ergodicity and Kolmogorov equations for dissipative SPDEs with singular drift: a variational approach

We prove existence of invariant measures for the Markovian semigroup generated by the solution to a parabolic semilinear stochastic PDE whose nonlinear drift term satisfies only a kind of symmetry condition on its behavior at infinity, but no restriction on its growth rate is imposed. Thanks to strong integrability properties of invariant measures $\mu$, solvability of the associated Kolmogorov equation in $L^1(\mu)$ is then established, and the infinitesimal generator of the transition semigroup is identified as the closure of the Kolmogorov operator. A key role is played by a generalized variational setting.Comment: 32 page

Fr\'echet differentiability of mild solutions to SPDEs with respect to the initial datum

We establish n-th order Fr\'echet differentiability with respect to the initial datum of mild solutions to a class of jump-diffusions in Hilbert spaces. In particular, the coefficients are Lipschitz continuous, but their derivatives of order higher than one can grow polynomially, and the (multiplicative) noise sources are a cylindrical Wiener process and a quasi-left-continuous integer-valued random measure. As preliminary steps, we prove well-posedness in the mild sense for this class of equations, as well as first-order G\^ateaux differentiability of their solutions with respect to the initial datum, extending previous results in several ways. The differentiability results obtained here are a fundamental step to construct classical solutions to non-local Kolmogorov equations with sufficiently regular coefficients by probabilistic means.Comment: 30 pages, no figure

From the viscous Cahn-Hilliard equation to a regularized forward-backward parabolic equation

A rigorous proof is given for the convergence of the solutions of a viscous Cahn-Hilliard system to the solution of the regularized version of the forward-backward parabolic equation, as the coefficient of the diffusive term goes to 0. Non-homogenous Neumann boundary condition are handled for the chemical potential and the subdifferential of a possible non-smooth double-well functional is considered in the equation. An error estimate for the difference of solutions is also proved in a suitable norm and with a specified rate of convergence.Comment: Key words and phrases: Cahn-Hilliard system, forward-backward parabolic equation, viscosity, initial-boundary value problem, asymptotic analysis, well-posednes